How can we use the patterns in a tree diagram to find the total number of branches?
10-108.
Nick came across the following problem: If a 4-digit number is randomly selected from all of
the 4-digit numbers that use the digits 1, 2, 3, 4, 5, 6, and 7, with repeated digits allowed, what is
the probability that the selected number is 2763? Nick knew that he had to figure out how many
numbers were possible, in order to know the size of his sample space.
a. Nick started to make a systematic list of the possibilities, but after the first few he gave up. What is the
difficulty in trying to create a list?
b. Next, he started a tree diagram. What problem did he encounter with the tree?
c. Nick decided he needed a shortcut strategy for organizing this problem; otherwise he was going to be up
all night. He started by asking himself, "How many decisions (about the digits) do I need to
make?” With your team discuss his first question and then consider his next question, "How many
choices do I have for each decision (each digit)?"
d. Audrey was at her house working on the same problem. She was thinking of a tree diagram, when she
asked herself, "How many branch points will this tree have?" and "How many branches at each
point?” What are the answers to her questions? How are these questions related to the one that Nick
was pondering?
e. At the same moment, they text messaged each other that they were stuck. When they talked, they
realized they were on the same track. The problem asks for four-digit numbers, so there are four
decisions. Simultaneously they said, "We need a decision chart." They wrote the following on their
papers:
1st digit 2nd digit 3rd digit 4th digit
How many choices are there for each decision? How many four-digit numbers are there?
f. What is the probability that the randomly selected number will be 2763?